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\newcommand{\CourseName}{复变函数作业1AB}
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%\section{2.1.1. Limits and Continuity. Example. } %%1
%
%\subsection{Example. }

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\section{The Algebra of Complex Numbers}

\begin{enumerate}%\itemsep1.8cm

\item  (***) %1. 
Find the values of
$$
(1+2i)^3, \hspace{0.5cm}
\frac{5}{-3+4i}, \hspace{0.5cm}
\left(\frac{2+i}{3-2i}\right)^2, \hspace{0.5cm}
(1+i)^n+(1-i)^n.
$$

\item (***)  %2. 
Compute
$$
\sqrt{i}, \hspace{0.5cm}
\sqrt{-i}, \hspace{0.5cm}
\sqrt{1+i}, \hspace{0.5cm}
\sqrt{\frac{1-i\sqrt{3}}{2}}.
$$

\item (***)  %3. 
Show that the system of all matrices of the special form 
$$
\begin{pmatrix} \alpha & \beta \\ -\beta & \alpha \end{pmatrix}
$$
combined by matrix addition and matrix multiplication, is isomorphic to
the field of complex numbers.


\item  %4. 
Prove Lagrange's identity in the complex form
$$
\left\vert \sum\limits_{i=1}^n a_ib_i \right\vert^2 
= \sum\limits_{i=1}^n |a_i|^2 \sum\limits_{i=1}^n |b_i|^2
- \sum\limits_{1\le i<j\le n} |a_i\bar{b_j}-a_j\bar{b_i}|^2. 
$$


\item  %5.
Prove that the sign of equality holds in the inequality 
\begin{equation*}
|a_1+a_2+\cdots+a_n| \le |a_1| + |a_2| + \cdots + |a_n|. 
%\label{eq-11}
\end{equation*}
if and only if the ratio of any two nonzero terms is positive. 


\item  %6.
Prove Cauchy's inequality in the complex form
\begin{equation*}
|a_1b_1+\cdots+a_nb_n|^2 \le (|a_1|^2+\cdots+|a_n|^2)(|b_1|^2+\cdots+|b_n|^2),
%\label{eq-7}
\end{equation*}


\item (***)  %7.  
Visualize the arithmetic operations of complex numbers in the the complex plane, including addition, subtraction, multiplication and division. 


\item (***)  %8. 
Find the symmetric points of the complex number $a$ with respect to the lines which bisect the angles between the coordinate axes.


\item  %9. 
Find the center and the radius of the circle which circumscribes the triangle with vertices $a_1, a_2, a_3$. Express the result in symmetric form.



\end{enumerate}

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\newpage
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\section{The Geometry Representation of Complex Numbers}

\begin{enumerate}%\itemsep1.8cm

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\item  %1. 
Prove that
$$
\left\vert \frac{a-b}{1-\bar{a}b}\right\vert = 1
$$
if either $|a| = 1$ or $|b| = 1$. 
What exception must be made if $|a| = |b| = 1$?

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\item (***)  %2. 
Prove that
$$
\left\vert \frac{a-b}{1-\bar{a}b} \right\vert < 1
$$
if $|a| < 1$ and $|b| < 1$.

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\item  %3. 
Express $\cos 3\varphi$, $\cos 4\varphi$, and $\sin 5\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$.

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\item (***)  %4. 
When does $az + b\bar{z} + c = 0$ represent a line?


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\item (***)  %5. 
Write the equation of an ellipse, hyperbola, parabola in complex form.


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\item  %6. 
Show that all circles that pass through $a$ and $1/\bar{a}$ intersect the circle $|z|=1$ at right angles.

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\item  %7. 
Prove that the distance between the stereographic projections of two complex numbers $z$ and $z'$ is given by
\begin{equation*}
d(z,z') = \frac{2|z-z'|}{\sqrt{(1+|z|^2)(1+|z'|^2)}}. 
%\label{eq-}
\end{equation*}

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\item (***)  %8. 
Show that two complex numbers $z$ and $z'$ correspond to diametrically opposite points on the Riemann sphere if and only if $zz' = -1$.

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\end{enumerate}

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\end{document}

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